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Properties

Label 1-431-431.108-r0-0-0
Degree $1$
Conductor $431$
Sign $0.798 - 0.602i$
Analytic cond. $2.00155$
Root an. cond. $2.00155$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.744 − 0.667i)2-s + (−0.997 + 0.0729i)3-s + (0.109 − 0.994i)4-s + (0.957 + 0.288i)5-s + (−0.694 + 0.719i)6-s + (0.989 − 0.145i)7-s + (−0.581 − 0.813i)8-s + (0.989 − 0.145i)9-s + (0.905 − 0.424i)10-s + (−0.181 + 0.983i)11-s + (−0.0365 + 0.999i)12-s + (0.744 + 0.667i)13-s + (0.639 − 0.768i)14-s + (−0.976 − 0.217i)15-s + (−0.976 − 0.217i)16-s + (−0.181 + 0.983i)17-s + ⋯
L(s)  = 1  + (0.744 − 0.667i)2-s + (−0.997 + 0.0729i)3-s + (0.109 − 0.994i)4-s + (0.957 + 0.288i)5-s + (−0.694 + 0.719i)6-s + (0.989 − 0.145i)7-s + (−0.581 − 0.813i)8-s + (0.989 − 0.145i)9-s + (0.905 − 0.424i)10-s + (−0.181 + 0.983i)11-s + (−0.0365 + 0.999i)12-s + (0.744 + 0.667i)13-s + (0.639 − 0.768i)14-s + (−0.976 − 0.217i)15-s + (−0.976 − 0.217i)16-s + (−0.181 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.798 - 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(431\)
Sign: $0.798 - 0.602i$
Analytic conductor: \(2.00155\)
Root analytic conductor: \(2.00155\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{431} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 431,\ (0:\ ),\ 0.798 - 0.602i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.834887354 - 0.6145403861i\)
\(L(\frac12)\) \(\approx\) \(1.834887354 - 0.6145403861i\)
\(L(1)\) \(\approx\) \(1.441865098 - 0.4258902440i\)
\(L(1)\) \(\approx\) \(1.441865098 - 0.4258902440i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad431 \( 1 \)
good2 \( 1 + (0.744 - 0.667i)T \)
3 \( 1 + (-0.997 + 0.0729i)T \)
5 \( 1 + (0.957 + 0.288i)T \)
7 \( 1 + (0.989 - 0.145i)T \)
11 \( 1 + (-0.181 + 0.983i)T \)
13 \( 1 + (0.744 + 0.667i)T \)
17 \( 1 + (-0.181 + 0.983i)T \)
19 \( 1 + (0.957 - 0.288i)T \)
23 \( 1 + (-0.872 + 0.489i)T \)
29 \( 1 + (0.109 + 0.994i)T \)
31 \( 1 + (-0.694 - 0.719i)T \)
37 \( 1 + (0.639 - 0.768i)T \)
41 \( 1 + (-0.976 - 0.217i)T \)
43 \( 1 + (0.905 + 0.424i)T \)
47 \( 1 + (-0.457 - 0.889i)T \)
53 \( 1 + (-0.457 + 0.889i)T \)
59 \( 1 + (0.744 - 0.667i)T \)
61 \( 1 + (0.252 - 0.967i)T \)
67 \( 1 + (0.639 - 0.768i)T \)
71 \( 1 + (-0.457 - 0.889i)T \)
73 \( 1 + (-0.791 - 0.611i)T \)
79 \( 1 + (-0.322 - 0.946i)T \)
83 \( 1 + (-0.997 + 0.0729i)T \)
89 \( 1 + (0.391 - 0.920i)T \)
97 \( 1 + (-0.322 + 0.946i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.29686318263268941913441179482, −23.47252221114362479858374124661, −22.40330946493396046145481918275, −21.938613750601649197826856852708, −20.96312043474157913165402141771, −20.533467857655768919510118348432, −18.40123883522972228634223910597, −18.0131777356562041466516586357, −17.22900606929802700842259284221, −16.271845349060229489319474953875, −15.782828599533481411839231582654, −14.39922000558181499728471424570, −13.67241485333596268062697944416, −12.95921577827638586453937267037, −11.82054005016081262359544034015, −11.219018179185230342702684515594, −10.0632851172316915353968535351, −8.67591415925421415456121035253, −7.80156453352524222383861931103, −6.59218167394768554928614127424, −5.613356103995580198155370934033, −5.33680272161260496315211840661, −4.19126854381940296556624548016, −2.681357033227719192063464567009, −1.19441352365180740109283408885, 1.47008941765102386861253601819, 1.97579233710681439094106413626, 3.76376630247808542781069629418, 4.74142975698060438655154308294, 5.53325003455571206816973359037, 6.35598306804412062526281048074, 7.378425298321807078244330139838, 9.20940012504188442234769910149, 10.10095302507820984145730331875, 10.873617017132091617060931992, 11.52880023691039588309601344652, 12.52438022279691714227368756724, 13.3735085191682137167725821875, 14.26686524964993001170746263995, 15.098672306180763558059855765890, 16.14847013973351704563135173392, 17.37191439889192692956101244380, 18.044512338875344457604929500662, 18.59788408688824437786017370767, 20.10323582030588689445061425122, 20.84362102476751412831075874467, 21.70758367614298484431147847606, 22.05020370416015140992504947606, 23.17279458295610461374845225929, 23.80304220998083646976038629164

Graph of the $Z$-function along the critical line